It is well known that the complexity, i.e. the number of vertices, edges and faces, of the 3-dimensional Voronoi diagram of n points can be as bad as Θ(n 2). It is also known that if the points are chosen Independently Identically Distributed uniformly from a 3-dimensional region such as a cube or sphere, then the expected complexity falls to O(n). In this paper we introduce the problem of analyzing what occurs if the points are chosen from a 2-dimensional region in 3-dimensional space. As an example, we examine the situation when the points are drawn from a Poisson distribution with rate n on the surface of a convex polytope. We prove that, in this case, the expected complexity of the resulting Voronoi diagram is O(n). © 2003 Elsevier Science B.V.
Golin, M. J., & Na, H. S. (2003). On the average complexity of 3D-Voronoi diagrams of random points on convex polytopes. Computational Geometry: Theory and Applications, 25(3), 197–231. https://doi.org/10.1016/S0925-7721(02)00123-2